The API unlock code provided to clients who purchase the API and the API serial number (provided to partners) both unlock the API. So, what is the difference?
Please review this KB: What is the difference between the API unlock code and the API serial number?

If a client has already purchased the API module and has it unlocked, is the serial number needed?
No. Since they both provide the same function, the serial number will not be needed. The drawback is that the API module will have to be unlocked on each machine running the API application. 

Is the API serial number the same as the Database serial number seen under Help > About? 
No. The API serial number is a unique 20-digit number that allows Blackbaud partners to create and implement a custom API application on a client's machine without having to purchase and unlock the API module.

How and where is this API serial number used?
Please review this KB: What changes need to be made to the API to allow a license free 3rd party integration?

Is the API serial number specific to the Blackbaud Technology partner requesting it?
No. The serial number is specific only to the client name provided in the request. It will not work for any other client, so each client will require their own unique serial number.

Would the use of this code be an additional “license” for the customer? 
Yes, the API does consume a total of one (1) user license regardless of whether an unlock code or serial number is used. 

Am I correct to assume that this serial number/code is for a free API license because we are a Blackbaud Technology Partner? 
Yes

How can I request an API serial number for a Blackbaud client?
If you're a registered Blackbaud Technology Partner or Solution Provider, you can create a case and request the API serial number with the exact name (case, spelling and punctuation must match) of the client seen in the Help > About screen.